About a week ago, I followed the AWS Route 53 instructions for creating a subdomain hosted zone to handle traffic for "request.i-codedesign.com".
The hosted zone provisioned these NS servers:
ns-1009.awsdns-62.net.
ns-1198.awsdns-21.org.
ns-1.awsdns-00.com.
ns-1807.awsdns-33.co.uk.
Also a week ago, I updated our DNS Service (Dotster) to add the above Name Server records: 
For a while this worked well, traffic was getting to the intended server.
Within the past week, it stopped working. What steps can I take to troubleshoot this? I've tried nslookup request.i-codedesign.com ns-1.awsdns-00.com, for example, but it just returns four sets of "DNS request timed out. timeout was 2 seconds."
EDIT
When I try from home, the address resolves and I can see my web app.
From here, this is the nslookup result:
C:\jhfrench>nslookup Default Server: Wireless_Broadband_Router.home Address: 192.168.1.1 > set debug > request.i-codedesign.com Server: Wireless_Broadband_Router.home Address: 192.168.1.1 ------------ Got answer: HEADER: opcode = QUERY, id = 2, rcode = NXDOMAIN header flags: response, auth. answer, want recursion, recursion avail. questions = 1, answers = 0, authority records = 1, additional = 0 QUESTIONS: request.i-codedesign.com.home, type = A, class = IN AUTHORITY RECORDS: -> home ttl = 3600 (1 hour) primary name server = home responsible mail addr = (root) serial = 10274 refresh = 3600 (1 hour) retry = 3600 (1 hour) expire = 3600 (1 hour) default TTL = 3600 (1 hour) ------------ ------------ Got answer: HEADER: opcode = QUERY, id = 3, rcode = NXDOMAIN header flags: response, auth. answer, want recursion, recursion avail. questions = 1, answers = 0, authority records = 1, additional = 0 QUESTIONS: request.i-codedesign.com.home, type = AAAA, class = IN AUTHORITY RECORDS: -> home ttl = 3600 (1 hour) primary name server = home responsible mail addr = (root) serial = 10275 refresh = 3600 (1 hour) retry = 3600 (1 hour) expire = 3600 (1 hour) default TTL = 3600 (1 hour) ------------ ------------ Got answer: HEADER: opcode = QUERY, id = 4, rcode = NOERROR header flags: response, want recursion, recursion avail. questions = 1, answers = 1, authority records = 0, additional = 0 QUESTIONS: request.i-codedesign.com, type = A, class = IN ANSWERS: -> request.i-codedesign.com internet address = 54.225.172.91 ttl = 300 (5 mins) ------------ Non-authoritative answer: ------------ Got answer: HEADER: opcode = QUERY, id = 5, rcode = NOERROR header flags: response, want recursion, recursion avail. questions = 1, answers = 0, authority records = 1, additional = 0 QUESTIONS: request.i-codedesign.com, type = AAAA, class = IN AUTHORITY RECORDS: -> request.i-codedesign.com ttl = 900 (15 mins) primary name server = ns-1828.awsdns-36.co.uk responsible mail addr = awsdns-hostmaster.amazon.com serial = 1 refresh = 7200 (2 hours) retry = 900 (15 mins) expire = 1209600 (14 days) default TTL = 86400 (1 day) ------------ Name: request.i-codedesign.com Address: 54.225.172.91